We develop a weak numerical Euler scheme for non-linear stochastic delay differential equations (SDDEs) driven by multidimensional Brownian motion. The weak Euler scheme has order of convergence 1, as in the case of stochastic ordinary differential equations (SODEs) (i.e., without delay). The result holds for SDDEs with multiple finite fixed delays in the drift and diffusion terms. Although the...

We are concerned with the exponential mean-square stability of two-step Maruyama methods for stochastic differential equations with time delay. We propose a family of schemes and prove that it can maintain the exponential mean-square stability of the linear stochastic delay differential equation for every step size of integral fraction of the delay in the equation. Numerical results for linear ...

Abstract: The current paper is concerned with the controllability of nonlocal secondorder impulsive neutral stochastic functional integro-differential equations with infinite delay and Poisson jumps in Hilbert spaces. Using the theory of a strongly continuous cosine family of bounded linear operators, stochastic analysis theory and with the help of the Banach fixed point theorem, we derive a ne...

An equivalence of the exponential stability concerning stochastic differential equations (SDEs), delay (SDDEs), and their corresponding Euler-Maruyama (EM) methods, is established. We show that for these four processes can be deduced from each other, provided or step size small enough. Using this relationship, we obtain between SDEs (or SDDEs) numerical methods difference) delay-free equations....

this paper presents an approach for solving a nonlinear stochastic differential equations (nsdes) using a new basis functions (nbfs). these functions and their operational matrices areused for representing matrix form of the nbfs. with using this method in combination with the collocation method, the nsdes are reduced a stochastic nonlinear system of equations and unknowns. then, the error anal...

In this paper, we discuss a new type of differential equations which we call anticipated backward stochastic differential equations (anticipated BSDEs). In these equations the generator includes not only the values of solutions of the present but also the future. We show that these anticipated BSDEs have unique solutions, a comparison theorem for their solutions, and a duality between them and ...

So far there are not many results on the attractor for the solutions of stochastic differential delay equations. The main aim of this paper is to establish new results on the attractor, from which follow several new criteria on the almost surely asymptotic stability for stochastic differential delay equations. As another by product, a number of new criteria on the boundedness of the solutions a...

-This paper deals with the adapted Milstein method for solving linear stochastic delay differential equations. It is proved that the numerical method is mean-square (MS) stable under suitable conditions. The obtained result shows that the method preserves the stability property of a class of linear constant-coefficient problems. This is also verified by several numerical examples. (~) 2006 Else...